4.9.A. The student is expected to demonstrate translations, reflections, and rotations using concrete models.
This lesson is an introduction to transformational geometry terms such
as translation, reflection, and rotations.
Point out different tessellating patterns which are evident in the environment - ceiling tiles, floor tiles, carpet designs, posters in classroom
To tessellate is to create a tiled design on a flat surface using a repeated geometric pattern without overlapping or leaving empty spaces.
Create a tessellation using pattern blocks
Using a orange square pattern block on the overhead, create a basic tessellation.
Ask students which of the other pattern block pieces they believe will tessellate
Do the same with the green triangle, blue rhombus, red trapezoid, and yellow hexagon.
All shapes will tessellate. The hexagon will leave spaces but the spaces repeat as well.
See the following sites:http://forum.swarthmore.edu/sum95/suzanne/active.html,
Introduce M.C. Escher
M.C. Escher is a Dutch artist whose tessellation drawings are largely based on recognizable images. He imaginatively turned geometric shapes into such creatures as birds, lizards, and fish. Most of his tessellation art was created from about 1937 - 1958. He did not consider himself a mathematician. In fact he had been a poor math student. His interest in tessellation art led him into the world of mathematics and he made significant contributions in both areas. For more about the life of M.C. Escher see the following site:http://www.WorldOfEscher.com/How To:
The "SNIP" Technique
Many geometric shapes tessellate. Polygons that tessellate can be altered to create irregularly shaped pictures. Using a simple "SNIP" technique, students can easily create irregularly shaped tiles that will become tessellating pictures.
The "SNIP" technique permits learners to change the shapes of figures. One overall rule to follow when altering a geometric shape to create a tessellation is that the shape must retain the same area as the original shape.
Translations or Slides
This transformation is restricted to polygons - parallelograms and hexagons whose opposite sides are parallel and congruent because an operation on one side always affects the opposite side.
- Use a 3" square, 3" rhombus, 2" x 4" rectangle or a hexagon each side being 2", copy and cut from heavy card stock or old manila folders one of the shapes for each student. All students start with the square or rectangle, which are some of the easiest shapes to use, and experiment with the other shapes later.
- One side of this piece should first be colored completely with crayon to prevent students from inadvertently flipping the piece while moving or taping their "SNIP"
- Demonstrate the "SNIP" technique on the overhead projector. For a beginner the easiest technique is to cut into one corner of the rectangle and ending on the adjacent corner. The SNIP in between can be curved or jagged. The simpler the better. A SNIP that is cut from adjacent corners will be easier to line up when matching side.
"Snip" from one corner to an adjacent one
- Warn students that when they cut their SNIP, no scraps may be discarded. Every piece has to be accounted for.
- Next demonstrate how to translate(slide) the newly cut SNIP across to the congruent and parallel side. It must match the straight edges and corners before being attached to the side. Tape the SNIP carefully and securely in its new home.
This movement is a slide transformation
- Since a rectangle has four sides, a second SNIP can be cut from one of the other pair of parallel sides and slide to the opposite side, once again matching the straight edges very carefully and then taping it into place. Remember no trimming to fit!
Sides can be altered in this manner
- When students have finished SNIPPING and taping the side of their rectangle, they are ready to tessellate with the resulting shape. I have the students take their shapes and place it on a 12 x 18 piece of construction paper. I put either double sided tape or sticky tack on one side of the shape to help hold the shape in place while they trace around the shape and this also keeps them from flipping the shape over.
- Stress to the students that they must be very careful in lining up their shape with the sides of the shapes they have already traced.
- Students complete their project by carefully adding detail and color to their tessellated design.
Rotations or turns
This transformation is restricted to polygons - triangles, parallelograms, and hexagons - with adjacent sides that are congruent.
- Begin with the parallelogram. Cut the shape from tag board or a manila folder and have them color one side to distinguish a front and a back. Using the snip technique, students cut from corner to corner of their parallelogram, but this time they rotate the snip at its endpoint to an adjacent side of their parallelogram, not an opposite side. Again they tape the piece securely into place after carefully matching the straight edges.
This transformation is a rotation
- The students alter another side of the parallelogram and rotate this snip from its endpoint to the adjacent side of the parallelogram and tape. I strongly recommend that all the sides of the parallelogram be altered differently to avoid confusion when tessellating.
All sides of the original shape have been altered
- Students now rotate or turn the shape as they move and trace the tile to cover the plane.
Rotation or turn at midpoints
See examples of student work:http://mathforum.com/alejandre/students.tess.html
- Take a triangle and students use a ruler to mark the midpoint of each of the three sides of the triangle
- Then they proceed to make a snip from one corner to the marked midpoint of a side of the triangle.
- They rotate this piece about the side's midpoint onto the remaining half of this same side, then tape.
- They repeat this procedure for all four sides of the triangle.
All sides of the original shape have been altered
Other great web sites about tesselations:Tantalizing Tesselations
How To Make A Tesselation
Celebrate 100 Years With M.C. Escher
Symmetry and Tesselations